# Math Help - Primitive Roots and Quadratic Residues

1. ## Primitive Roots and Quadratic Residues

I'm struggling with the concept of primitive roots and their application in certain proofs. I'm struggling with starting this problem:
Let a and n be in the natural numbers and let p be an odd prime where p does not divide a. Using primitive roots show $x^2 \equiv a$ mod p is solvable if and only if $y^2 \equiv a$ mod p^n is solvable.
2. The idea is that $\alpha,...,\alpha^{p-1}$ are a complete set of residues mod p when $\alpha$ is a primitive root.
So you can write $a\equiv\alpha^k$ for some integer $k$, given any nonzero a.
You might want to show that $a^{(p-1)/2}\equiv \pm 1$, the 1 and the -1 holding respectively if $a$ is a square or not.
To do this, assume $a$ is a power of some primitive root and see where you can take it from there.